Coherent Moving States in Highway Traffic (Originally: Moving Like a Solid Block)

17 Pages·0.16 MB·English

by  Dirk Helbing

Coherent Moving States in Highway Traffic
9
9
9
1 Dirk Helbing∗ and Bernardo A. Huberman+

n
a
J ∗ II. Institute of Theoretical Physics, University of Stuttgart,

3
1
Pfaffenwaldring 57/III, 70550 Stuttgart, Germany

]
h + Xerox PARC, 3333 Coyote Hill Road, Palo Alto, CA 94304, USA
c
e
[email protected], [email protected]
m
-
t
a
st Recent advances in multiagent simulations1−3 have made possible
.
t
a
the study of realistic traffic patterns and allow to test theories based
m
d- on driver behaviour3−6. Such simulations also display various empir-
n
o ical features of traffic flows7, and are used to design traffic controls
c
[
that maximise the throughput of vehicles in heavily transited high-

2
v
ways. In addition to its intrinsic economic value8, vehicular traffic
5
2
0 is of interest because it may throw light on some social phenomena
5
0 where diverse individuals competitively try to maximise their own
8
9
utilities under certain constraints9,10.
/
t
a
m
In this paper, we present simulation results that point to the exis-
-
d
n tence of cooperative, coherent states arising from competitive inter-
o
c
actions that lead to a new phenomenon in heterogeneous highway
:
v
i
X traffic. As the density of vehicles increases, their interactions cause
r
a a transition into a highly correlated state in which all vehicles prac-
tically move with the same speed, analogous to the motion of a solid
block. This state is associated with a reduced lane changing rate
and a safe, high and stable flow. It disappears as the vehicle density
exceeds a critical value. The effect is observed in recent evaluations
of Dutch traffic data.
1
Helbing/Huberman: Moving Like a Solid Block 2
In many social situations, decisions made by individuals lead to externalities,
which may be very regular, even without a globalcoordinator. In traffic, these
decisions concern when to accelerate or brake, pass, or enter into a heavily
transited multi-lane road11−14, while trying to get ahead as fast as possible,
but safely, under the constraints imposed by physical limitations and traffic
rules. At times this behaviorial rules give rise to very regular traffic patterns
as exemplified by the universal characteristics of moving jam fronts15 or syn-
chronised congested traffic16,17. These phenomena are in contrast with usual
social dilemmas, where cooperation in order to achieve a desirable collective
behaviour hinges on having small groups or long time horizons18,9.
In what follows we exhibit a new type of collective behaviour that we discov-
ered when studying the dynamics of a diverse set of vehicles, such as cars and
lorries, travelling through a two-lane highway with different velocities. As the
density of vehicles in the road increases, there is a transition into a highly
coherent state characterised by all vehicles having the same average velocity
and a very small dispersion around its value. The transition to this behavior
becomes apparent when looking at the travel time distributions of cars and
lorries, comprising the overall dynamics on a freeway stretch (Fig. 1). These
were obtained by running computer simulations using a discretised follow-
the-leader algorithm19,20, which distinguishes two neighbouring lanes i of an
unidirectional freeway. Both are subdivided into sites z ∈ {1,2,...,L} of
equal length ∆x = 2.5m. Each site is either empty or occupied, the latter
case representing the back of a vehicle of type a (e.g. a car or lorry) with
velocity v = u∆x/∆t. Here, u ∈ {0,1,...,umax} is the number of sites that
a
the vehicle moves per update step ∆t = 1s. Cars and lorries are characterised
Helbing/Huberman: Moving Like a Solid Block 3
by different ‘optimal’ or ‘desired’ (i.e. maximally safe) velocities U (d ) with
a +
which the vehicles would like to drive at a distance d to the vehicle in front
+
(see symbols inFig.2). Their lengths l correspond to themaximum distances
a
satisfying U (l ) = 0. At times T ∈ {1,2,...}, i.e. every time step ∆t, the
a a
positions z(T), velocities u(T), and lanes i(T) of all vehicles are updated in
parallel. Wehaveruledoutsynchronisationartefacts22 bythisupdatemethod,
which is appropriate for flow simulations1.
Denoting the position, velocity, and distance of the respective leading vehicle
(+) or following vehicle (−) on lane i(T) by z±, u±, and d± = |z±−z|, in the
′ ′ ′ ′
adjacent lane by z , u , and d = |z −z|, the successive update steps are:
± ± ± ±
1. Determine the potential velocities u(T + 1) and u′(T + 1) on the present
and the adjacent lane according to the acceleration law21
u(′)(T +1) = λU (d(′)(T))+(1−λ)u(T) ,
j a + k
where the floor function ⌊x⌋ is defined by the largest integer l ≤ x. This de-
scribes the typical follow-the-leader behaviour of driver-vehicle units. Delayed
by the reaction time ∆t, they tend to move with their desired velocity U , but
a
the adaptation takes a certain time τ = λ∆t because of the vehicle’s inertia.
2. Change lane in accordance with, for simplicity, symmetrical (‘American’)
rules, if the following incentive and safety criteria14 are fulfilled: Check if the
′
distanced (T)tothefollowingvehicle intheneighbouring laneisgreaterthan
−
′
thedistanceu (T+1)thatthisvehicle isexpected tomovewithinthereaction
−
′ ′
time ∆t (safety criterion 1). If so, the difference D = d (T)−u (T +1) > 0
− −
defines the backward surplus gap. Next, look if you could go faster in the
adjacentlane(incentivecriterion). Finally,makesurethattherelativevelocity
Helbing/Huberman: Moving Like a Solid Block 4
′ ′
[u (T +1)−u(T +1)] would not be larger than D +m (safety criterion 2),
−
where the magnitude of the parameter m ≥ 0 is a measure of how aggressive
drivers are in overtaking (by possibly enforcing deceleration manoeuvres).
3.If, intheupdatedlanei(T+1), thecorrespondingpotentialvelocity u(T+1)
′
or u(T +1) is positive, diminish it by 1 with probability p in order to account
for delayed adaptation and the variation of vehicle velocities.
4. Update the vehicle position according to the equation of motion z(T +1) =
z(T)+u(T +1).
Despite its simplifications, this model is in good agreement with the empirical
known features of traffic flows, and it can be well calibrated21: λ and U (d )
a +
determine theapproximatevelocity-density relationandtheinstability region.
Thetypical outflowfromtrafficjamsandtheircharacteristic dissolutionveloc-
ity15 can be enforced by ∆t and ∆x. The average distance between successive
traffic jams increases with smaller p. m allows to calibrate the lane-changing
rates. Our simulations started with uniform distances among the vehicles and
their associated desired velocities. The lorries were randomly selected. Since
our evaluations started after a transient period of one hour and extended over
anotherfourhours, theresultsarelargelyindependent oftheinitialconditions.
Investigating thedensity-dependent averagevelocities ofcarsandlorriesyields
further insight into the solid-like state (Fig. 2). For small p one finds that,
at a certain ‘critical’ density, the average speed of cars decays significantly
towards the speed of the lorries, which is still close to their maximum veloc-
ity. At this density, the freeway space is almost used up by the safe vehicle
headways, so that sufficiently large gaps for lane-changing can only occur for
Helbing/Huberman: Moving Like a Solid Block 5
strongly varying vehicle velocities (like for large p). However, since the speeds
ofcars andlorries arealmost identical inthe solid-like state, thelane-changing
rate drops by almost one order of magnitude (Fig. 3). Consequently, with-
out opportunities for overtaking, all vehicles have to move coherently at the
speed of the lorries, which closes the feedback-loop that causes the transition.
The solid-like flow does not change by adding vehicles until the whole freeway
is saturated by the vehicular space requirements at the speed of the lorries.
Then, the vehicle speeds decay significantly to maintain safe headways. The
onset of stop-and-go traffic at this density produces largely varying gaps, so
that overtaking is againpossible and the coherent state is destroyed. For large
p, we do not have a breakdown of the lane-changing rate at the critical density
and, hence, no coherent state. Nevertheless, lane-changing cars begin to in-
terfer with the lorries, so that the average velocity of lorries starts to decrease
with growing density before the average car velocity comes particularly close
to it (Fig. 2c).
As shown in Figure 4, our prediction of the transition into a coherent state is
supported by empirical data obtained fromhighway traffic in the Netherlands.
Clearly, the difference between the average velocities of cars and lorries shows
the predicted minimum at a density around 25 vehicles per kilometer and
lane, where the average car velocity approaches the constant velocity of the
lorries (Fig. 4a,b). The fact that the empirically observed minimum is less
distinct than in Fig. 2a can be reproduced by higher values of the fluctuation
parameter (p ≈ 0.15) and points to a noisy transition. This interpretation is
also supported by the relative fluctuation of vehicle speeds (Fig. 4c), which
shows a minimum at the same density, while remaining finite. A similar result
Helbing/Huberman: Moving Like a Solid Block 6
is obtained for the interation rates of vehicles (Fig. 4d).
In summary, we have presented a novel effect in highway traffic that con-
sists in the formation of coherent motion out of a disorganised vehicle flow by
competitive interactions. The predicted solid-like state is supported by real
highway data, and our interpretation of the effect suggests that it is largely in-
dependent of the chosen driver-vehicle model (although the transition may be
less sharp in a continuous model). It would be interesting to see if the sponta-
neous appearance of coherent states is also found in other social or biological
systems, such as pedestrian crowds24, cell colonies25, or animal swarms26.
We conclude by noting that the coherent state of vehicle motion considerably
reduces the main sources of highway accidents: differences in vehicle speeds
and lane-changes27. It is also associated with maximum throughput in the
highway and located just before the transition to unstable traffic flow. Thus,
at a practical level it is desirable to implement traffic rules and design high-
way controls that will lead to traffic moving like a solid block. Close to the
transition point the formation of this coherent state could be supported by
traffic-dependent lane-changing restrictions and variable speed limits or by
automatic vehicle control systems. Compared to American (symmetric) lane-
changing rules, European ones seem to be less efficient: An asymmetric lane
usage, where lorries mainly keep on one lane and overtaking is carried out on
the other lane(s), motivates car drivers to avoid the lorry lane, so that the
effective freeway capacity is reduced up to 25%.
Helbing/Huberman: Moving Like a Solid Block 7
REFERENCES
1 Schreckenberg, M., Schadschneider, A., Nagel, K. & Ito, N. Discrete
stochastic models for traffic flow. Phys. Rev. E 51, 2939–2949 (1995).
2 Faieta, B. & Huberman, B. A. Firefly: A synchronization strategy for
urban traffic control (Xerox PARC, Palo Alto, Internal ReportNo. SSL-
42, 1993).
3 Schreckenberg, M. & Wolf, D. E. Traffic and Granular Flow ’97
(Springer, Singapore, 1998).
4 Prigogine,I.&Herman,R.Kinetic Theoryof VehicularTraffic(Elsevier,
New York, 1971).
5 Leutzbach, W. Introduction to the Theory of Traffic Flow (Springer,
Berlin, 1988).
6 Helbing, D. Verkehrsdynamik (Springer, Berlin, 1997).
7 May, A. D. Traffic Flow Fundamentals (Prentice Hall, Englewood Cliffs,
NJ, 1990).
8 Small, K. A. Urban Transportation Economics (Harwood Academic,
London, 1992).
9 Glance, N. S. & Huberman, B. A. The dynamics of social dilemmas.
Sci. Am. 270, 76–81 (1994).
10 Helbing, D. Quantitative Sociodynamics (Kluwer Academic, Dordrecht,
1995).
11 Daganzo, C. F. Probabilistic structure of two-lane road traffic. Transpn.
Res. 9, 339–346 (1975).
12 Hall, F. L. & Lam, T. N. The characteristics of congested flow on a free-
way across lanes, space, and time. Transpn. Res. A 22, 45–56 (1988).
Helbing/Huberman: Moving Like a Solid Block 8
13 Rickert, M., Nagel, K., Schreckenberg, M. & Latour, A. Two lane traffic
simulations using cellular automata. Physica A 231, 534–550 (1996).
14 Nagel, K., Wolf, D.E., Wagner, P. & Simon, P. Two-lane traffic rules for
cellular automata: A systematic approach. Phys. Rev. E 58, 1425–1437
(1998).
15 Kerner, B. S. & Rehborn, H. Experimental features and characteristics
of traffic jams. Phys. Rev. E 53, R1297–R1300 (1996).
16 Kerner, B. S. & Rehborn, H. Experimental properties of phase transi-
tions in traffic flow. Phys. Rev. Lett. 79, 4030–4033 (1997).
17 Helbing, D. & Treiber, M. Gas-kinetic-based traffic model explaining
observed hysteretic phase transition. Phys. Rev. Lett. 81, 3042–3045
(1998).
18 Axelrod, R. & Dion, D. The further evolution of cooperation. Science
242, 1385–1390 (1988).
19 Gazis, D. C., Herman, R. & Rothery, R. W. Nonlinear follow the leader
models of traffic flow. Operations Res. 9, 545–567 (1961).
20 Bando, M. et al. Phenomenological study of dynamical model of traffic
flow. J. Phys. I France 5, 1389–1399 (1995).
21 Helbing, D. Cellular automaton model simulating experimental proper-
ties of traffic flows. Submitted to Phys. Rev. (1998).
22 Huberman, B. A. & Glance, N. S. Evolutionary games and computer
simulations. Proc. Natl. Acad. Sci. USA 90, 7716–7718 (1993).
23 Krug, J. & Ferrari, P. A. Phase transitions in driven diffusive systems
with random rates, J. Phys. A: Math. Gen. 29, L465–L471 (1996).
24 Helbing, D., Keltsch, J. & Moln´ar, P. Modelling the evolution of human
trail systems. Nature 388, 47–50 (1997).
Helbing/Huberman: Moving Like a Solid Block 9
25 Ben-Jacob, E. et al. Generic modelling of cooperative growth patterns
in bacterial colonies. Nature 368, 46–49 (1994).
26 Vicsek, T. et al. Novel type of phase transition in a system of self-driven
particles. Phys. Rev. Lett. 75, 1226–1229 (1995).
27 Lave, C. A. Speeding, coordination, and the 55 MPH limit. American
Economic Review 75, 1159–1164 (1985).
Acknowledgments: D.H. wants to thank the DFG for financial support by
a Heisenberg scholarship. He is also grateful to Henk Taale and the Dutch
Ministry of Transport, Public Works and Water Management for supplying
the freeway data, which were evaluated by Vladimir Shvetsov.
Helbing/Huberman: Moving Like a Solid Block 10
FIGURES
Density: 5 veh/(km lane) Density: 21.3 veh/(km lane)
n n
o 1 o 1
ti Cars ti Cars
u u
b 0.8 Lorries b 0.8 Lorries
ri ri
t t
s s
Di 0.6 Di 0.6

e e
m 0.4 m 0.4
Ti Ti
0.2 0.2
el a el b
v v
a 0 a 0
r r
T 0 2 4 6 8 10 T 0 2 4 6 8 10
Travel Time (min) Travel Time (min)
Density: 21.6 veh/(km lane) Density: 25 veh/(km lane)
n n
o 1 o 1
ti Cars ti Cars
u u
b 0.8 Lorries b 0.8 Lorries
ri ri
t t
s s
Di 0.6 Di 0.6

e e
m 0.4 m 0.4
Ti Ti
0.2 0.2
el c el d
v v
a 0 a 0
r r
T 0 2 4 6 8 10 T 0 2 4 6 8 10
Travel Time (min) Travel Time (min)